| # | original ode | \(y=xf\left ( p\right ) +g\left ( p\right ) \) | \(f\left ( p\right ) \) | \(g\left ( p\right ) \) | type |
| \(1\) | \(x\left ( y^{\prime }\right ) ^{2}-yy^{\prime }=-1\) | \(y=xp+\frac {1}{p}\) | \(p\) | \(\frac {1}{p}\) | Clairaut |
| \(2\) | \(y=xy^{\prime }-\left ( y^{\prime }\right ) ^{2}\) | \(y=xp-p^{2}\) | \(p\) | \(-p^{2}\) | Clairaut |
| \(3\) | \(y=xy^{\prime }-\frac {1}{4}\left ( y^{\prime }\right ) ^{2}\) | \(y=xp-\frac {1}{4}p^{2}\) | \(p\) | \(-\frac {1}{4}p^{2}\) | Clairaut |
| \(4\) | \(y=x\left ( y^{\prime }\right ) ^{2}\) | \(y=xp^{2}\) | \(p^{2}\) | \(0\) | d’Alembert |
| \(5\) | \(y=x+\left ( y^{\prime }\right ) ^{2}\) | \(y=x+p^{2}\) | \(1\) | \(p^{2}\) | d’Alembert |
| \(6\) | \(\left ( y^{\prime }\right ) ^{2}-1-x-y=0\) | \(y=-x+\left ( p^{2}-1\right ) \) | \(-1\) | \(\left ( p^{2}-1\right ) \) | d’Alembert |
| \(7\) | \(yy^{\prime }-\left ( y^{\prime }\right ) ^{2}=x\) | \(y=\frac {1}{p}x+p\) | \(\frac {1}{p}\) | \(p\) | d’Alembert |
| \(8\) | \(y=x\left ( y^{\prime }\right ) ^{2}+\left ( y^{\prime }\right ) ^{2}\) | \(y=xp^{2}+p^{2}\) | \(p^{2}\) | \(p^{2}\) | d’Alembert |
| \(9\) | \(y=\frac {x}{a}y^{\prime }+\frac {b}{ay^{\prime }}\) | \(y=\frac {x}{a}p+\frac {b}{a}p^{-1}\) | \(\frac {p}{a}\) | \(\frac {b}{a}p^{-1}\) | d’Alembert |
| \(10\) | \(y=x\left ( y^{\prime }+a\sqrt {1+\left ( y^{\prime }\right ) ^{2}}\right ) \) | \(y=x\left ( p+a\sqrt {1+p^{2}}\right ) \) | \(p+a\sqrt {1+p^{2}}\) | \(0\) | d’Alembert |
| \(11\) | \(y=x+\left ( y^{\prime }\right ) ^{2}\left ( 1-\frac {2}{3}y^{\prime }\right ) \) | \(y=x+p^{2}\left ( 1-\frac {2}{3}p\right ) \) | \(1\) | \(p^{2}\left ( 1-\frac {2}{3}p\right ) \) | d’Alembert |
| \(12\) | \(y=2x-\frac {1}{2}\ln \left ( \frac {\left ( y^{\prime }\right ) ^{2}}{y^{\prime }-1}\right ) \) | \(y=2x-\frac {1}{2}\ln \left ( \frac {p^{2}}{p-1}\right ) \) | \(2\) | \(-\frac {1}{2}\ln \left ( \frac {p^{2}}{p-1}\right ) \) | d’Alembert |
| \(13\) | \(\left ( y^{\prime }\right ) ^{2}-x\left ( y^{\prime }\right ) ^{2}+y\left ( 1+y^{\prime }\right ) -xy^{\prime }=0\) | \(y=\frac {xp+xp^{2}-p^{2}}{p+1}=xp-\frac {p^{2}}{p+1}\) | \(p\) | \(-\frac {p^{2}}{p+1}\) | Clairaut |
| \(14\) | \(x\left ( y^{\prime }\right ) ^{2}+\left ( x-y\right ) y^{\prime }+1-y=0\) | \(y=xp+\frac {1}{1+p}\) | \(p\) | \(\frac {1}{1+p}\) | Clairaut |
| \(15\) | \(xyy^{\prime }=y^{2}+x\sqrt {4x^{2}+y^{2}}\) | \(y=\operatorname {RootOf}\left ( h(p)\right ) x\) | \(\operatorname {RootOf}\left ( h(p)\right ) \) | \(0\) | d’Alembert |
| \(16\) | \(\ln \left ( \cos y^{\prime }\right ) +y^{\prime }\tan y^{\prime }=y\) | \(y=\ln \left ( \cos y^{\prime }\right ) +y^{\prime }\tan y^{\prime }\) | \(0\) | \(\ln \left ( \cos p\right ) +p\tan p\) | d’Alembert |
| \(17\) | \(x\left ( y^{\prime }\right ) ^{2}-2yy^{\prime }+4x=0\) | \(y=x\left ( \frac {1}{2}y^{\prime }+2\frac {1}{y^{\prime }}\right ) \) | \(\frac {1}{2}p+\frac {2}{p}\) | \(0\) | d’Alembert |